32000 in Binary

The process of converting 32000 from decimal to binary involves dividing the number 32000 by 2. Here, it is getting divided by 2 because the binary number system uses only 2 digits (0 and 1).

The quotient becomes the dividend in the next step, and the process continues until the quotient becomes 0. This is a commonly used method to convert 32000 to binary. In the last step, the remainder is noted down bottom side up, and that becomes the converted value.

For example, the remainders noted down after dividing 32000 by 2 until getting 0 as the quotient is 1111101000000000. Remember, the remainders here have been written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 1111101000000000.

The second column represents the place values of each digit, and the third column is the value calculation, where the binary digits are multiplied by their corresponding place values.

The results of the third column can be added to cross-check if 1111101000000000 in binary is indeed 32000 in the decimal number system.

32000 can be converted easily from decimal to binary. The methods mentioned below will help us convert the number. Let’s see how it is done.

Expansion Method: Let us see the step-by-step process of converting 32000 using the expansion method.

Step 1 - Figure out the place values: In the binary system, each place value is a power of 2. Therefore, in the first step, we will ascertain the powers of 2. 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 ... 2^15 = 32768 Since 32768 is greater than 32000, we stop at 2^14 = 16384.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^14 = 16384. This is because, in this step, we have to identify the largest power of 2, which is less than or equal to the given number, 32000. Since 2^14 is the number we are looking for, write 1 in the 2^14 place. Now the value of 2^14, which is 16384, is subtracted from 32000. 32000 - 16384 = 15616.

Step 3 - Identify the next largest power of 2: In this step, we need to find the largest power of 2 that fits into the result of the previous step, 15616. So, the next largest power of 2 is 2^13, which is less than or equal to 15616. Now, we have to write 1 in the 2^13 place. And then subtract 8192 from 15616. 15616 - 8192 = 7424. Continue this process until there is no remainder. Finally, write the values in reverse order: Therefore, 1111101000000000 is 32000 in binary.

Grouping Method: In this method, we divide the number 32000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 32000 by 2. 32000 / 2 = 16000. Here, 16000 is the quotient and 0 is the remainder. Continue dividing the quotient by 2 until the quotient becomes 0.

Step 2 - Write down the remainders from bottom to top. Therefore, 32000 (decimal) = 1111101000000000 (binary).

There are certain rules to follow when converting any number to binary. Some of them are mentioned below:

Rule 1: Place Value Method

This is one of the most commonly used rules to convert any number to binary. The place value method is the same as the expansion method, where we need to find the largest power of 2. Let’s see a brief step-by-step explanation to understand the first rule. Find the largest power of 2 less than or equal to 32000. Since the answer is 2^14, write 1 next to this power of 2. Subtract the value (16384) from 32000. So, 32000 - 16384 = 15616. Find the largest power of 2 less than or equal to 15616. The answer is 2^13. So, write 1 next to this power. Continue this process, writing 1s and 0s in the appropriate places, until you have converted the entire number. Final conversion will be 1111101000000000.

Rule 2: Division by 2 Method

The division by 2 method is the same as the grouping method. A brief step-by-step explanation is given below for better understanding. First, 32000 is divided by 2 to get 16000 as the quotient and 0 as the remainder. Continue dividing the quotient by 2 until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 32000, 1111101000000000.

Rule 3: Representation Method

This rule also involves breaking the number into powers of 2. Identify the powers of 2 and write it down in decreasing order. Find the largest power that fits into 32000. Repeat the process and allocate 1s and 0s to the suitable powers of 2. Combine the digits (0 and 1) to get the binary result.

Rule 4: Limitation Rule

The limitation of the binary system is that only 0s and 1s can be used to represent numbers. The system doesn’t use any other digits other than 0 and 1. This is a base 2 number system, where the binary places represent powers of 2. So, every digit is either a 0 or a 1. To convert 32000, we use 1s and 0s as needed for each power of 2.

Learning a few tips and tricks is a great way to solve any mathematical problems easily. Let us take a look at some tips and tricks for binary numbers up to 32000.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers and understand patterns for larger ones.

Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary. 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 8 + 8 = 16 → 10000

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 32000 is even, and its binary form is 1111101000000000. Here, the binary of 32000 ends in 0. If the number is odd, then its binary equivalent will end in 1.

Cross-verify the answers: Once the conversion is done, we can cross-verify the answers by converting the number back to the decimal form. This will eliminate any unforeseen errors in conversion.

Practice by using tables: Writing the decimal numbers and their binary equivalents on a table will help us remember the conversions.

• Decimal: It is the base 10 number system which uses digits from 0 to 9.

• Binary: This number system uses only 0 and 1. It is also called the base 2 number system.

• Place value: Every digit has a value based on its position in a given number. For example, in 102 (base 10), 1 has occupied the hundreds place, 0 is in the tens place, and 2 is in the ones place.

• Octal: It is the number system with a base of 8. It uses digits from 0 to 7.

• Remainder: The number left after division. In binary conversion, remainders are used to construct the binary equivalent.